Analysis of nonlinear integral equations with Erdélyi–Kober fractional operator

This paper initiates the investigation of nonlinear integral equations with Erdélyi–Kober fractional operator. Existence and uniqueness results of solutions in a closed ball are obtained by using a directly computational method and Schauder fixed point theorem via a weakly singular integral inequality due to Ma and Pec˘arić [20]. Meanwhile, three certain solutions sets YK,σ, Y1,λ and Y1,1, which tending to zero at an appropriate rate t−ν, 0 < ν = σ (or λ or 1) as t → +∞, are constructed and local stability results of solutions are obtained based on these sets respectively under some suitable conditions. Two examples are given to illustrate the results.

► Nonlinear integral equations involving in Erdélyi–Kober fractional operator are studied.
► Weakly singular integral inequalities are widely used.
► Novel solutions sets tending to zero at an appropriate rate are introduced.